In summary, we analyze the IceCube data in four years from 2010 to 2014 and get a new constraint on the GRB neutrino flux. Among all 24 “shower” neutrino events above 60 TeV, 12 events are emitted with associated GRBs
by considering the Lorentz violation of neutrinos [1]}, [2]}.
In this paper we make an estimate of the background of our previous analyzed
events and show that they are well stand out beyond the backgrounds. Further more,
we find that “track" events can also be associated with GRBs
under the same Lorentz violation features of neutrinos.
Our results of track events are consistent with the shower events with a same Lorentz violation scale \(E_{\rm LV}\) . These results can be considered as new supports to the obtained LV scale \(E_{\rm LV}\)  from
previous works [3]}, [1]}, [2]}. It is also indicated that the neutrino flux from GRBs associated by us is comparable with the prediction from GRB fireball models. We therefore conclude that gamma ray bursts can serve as a significant source
for the ultra-high energy IceCube neutrino events
provided that there are Lorentz violation of neutrinos and anti-neutrinos.
Our work supports the Lorentz violation and \(CPT\) -violation of neutrinos as proposed in ref. [2]}, indicating new physics beyond relativity.
